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| Mirrors > Home > MPE Home > Th. List > nn0ssz | Structured version Visualization version GIF version | ||
| Description: Nonnegative integers are a subset of the integers. (Contributed by NM, 9-May-2004.) | 
| Ref | Expression | 
|---|---|
| nn0ssz | ⊢ ℕ0 ⊆ ℤ | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-n0 12529 | . 2 ⊢ ℕ0 = (ℕ ∪ {0}) | |
| 2 | nnssz 12637 | . . 3 ⊢ ℕ ⊆ ℤ | |
| 3 | 0z 12626 | . . . 4 ⊢ 0 ∈ ℤ | |
| 4 | c0ex 11256 | . . . . 5 ⊢ 0 ∈ V | |
| 5 | 4 | snss 4784 | . . . 4 ⊢ (0 ∈ ℤ ↔ {0} ⊆ ℤ) | 
| 6 | 3, 5 | mpbi 230 | . . 3 ⊢ {0} ⊆ ℤ | 
| 7 | 2, 6 | unssi 4190 | . 2 ⊢ (ℕ ∪ {0}) ⊆ ℤ | 
| 8 | 1, 7 | eqsstri 4029 | 1 ⊢ ℕ0 ⊆ ℤ | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∈ wcel 2107 ∪ cun 3948 ⊆ wss 3950 {csn 4625 0cc0 11156 ℕcn 12267 ℕ0cn0 12528 ℤcz 12615 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-un 7756 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-i2m1 11224 ax-1ne0 11225 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-om 7889 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-neg 11496 df-nn 12268 df-n0 12529 df-z 12616 | 
| This theorem is referenced by: nn0z 12640 nn0zd 12641 nn0zi 12644 nn0ssq 13000 nthruz 16290 oddnn02np1 16386 evennn02n 16388 bitsf1ocnv 16482 pclem 16877 0ram 17059 0ram2 17060 0ramcl 17062 gexex 19872 iscmet3lem3 25325 plyeq0lem 26250 dgrlem 26269 2sqreultblem 27493 archirngz 33197 dffltz 42649 diophrw 42775 diophin 42788 diophun 42789 eq0rabdioph 42792 eqrabdioph 42793 rabdiophlem1 42817 diophren 42829 etransclem48 46302 | 
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